Loose cores and cycles in random hypergraphs

article Loose cores and cycles in random hypergraphs published yes Oliver Cooley author 43f4ddd0-a46b-11ec-8df6-ef3703bd721d Mihyun Kang author Julian Zalla author MaKw department Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structurewhich mirrors the close relationship between cycles and $2$-cores in graphs. We prove that in the $r$-uniform binomial random hypergraph $H^r(n,p)$, the order of the loose core undergoes a phase transition at a certain critical threshold and determine this order, as well as the number of edges, asymptotically in the subcritical and supercritical regimes. Our main tool is an algorithm called CoreConstruct, which enables us to analyse a peeling process for the loose core. By analysing this algorithm we determine the asymptotic degree distribution of vertices in the loose core and in particular how many vertices and edges the loose core contains. As a corollary we obtain an improved upper bound on the length of the longest loose cycle in $H^r(n,p)$. https://research-explorer.ista.ac.at/download/12286/12462/2022_ElecJournCombinatorics_Cooley_Kang_Zalla.pdf application/pdfno https://creativecommons.org/licenses/by-nd/4.0/ The Electronic Journal of Combinatorics2022 eng Computational Theory and MathematicsGeometry and TopologyTheoretical Computer ScienceApplied MathematicsDiscrete Mathematics and Combinatorics The Electronic Journal of Combinatorics 1077-8926 00087676330000110.37236/10794 294 O. Cooley, M. Kang, and J. Zalla, “Loose cores and cycles in random hypergraphs,” The Electronic Journal of Combinatorics, vol. 29, no. 4. The Electronic Journal of Combinatorics, 2022. O. Cooley, M. Kang, J. Zalla, The Electronic Journal of Combinatorics 29 (2022). Cooley, O., Kang, M., & Zalla, J. (2022). Loose cores and cycles in random hypergraphs. The Electronic Journal of Combinatorics. The Electronic Journal of Combinatorics. <a href="https://doi.org/10.37236/10794">https://doi.org/10.37236/10794</a> Cooley, Oliver, et al. “Loose Cores and Cycles in Random Hypergraphs.” The Electronic Journal of Combinatorics, vol. 29, no. 4, P4.13, The Electronic Journal of Combinatorics, 2022, doi:<a href="https://doi.org/10.37236/10794">10.37236/10794</a>. Cooley O, Kang M, Zalla J. Loose cores and cycles in random hypergraphs. The Electronic Journal of Combinatorics. 2022;29(4). doi:<a href="https://doi.org/10.37236/10794">10.37236/10794</a> Cooley O, Kang M, Zalla J. 2022. Loose cores and cycles in random hypergraphs. The Electronic Journal of Combinatorics. 29(4), P4.13. Cooley, Oliver, Mihyun Kang, and Julian Zalla. “Loose Cores and Cycles in Random Hypergraphs.” The Electronic Journal of Combinatorics. The Electronic Journal of Combinatorics, 2022. <a href="https://doi.org/10.37236/10794">https://doi.org/10.37236/10794</a>. 122862023-01-16T10:03:57Z2023-08-04T10:29:18Z